Combine the berries, yogurt, sugar, and lemon juice in a blender and purée until smooth.
Pour the mixture through a strainer and use a rubber spatula to help pass the liquid through, removing any seeds and skin from the berries.
Chill the yogurt mixture in the fridge for at least one hour, but up to over night.
Pour into your ice cream maker and churn according to the manufacturer's directions — around 30 minutes.
Remove the frozen yogurt from the ice cream maker into a freezer storage container and freeze until firm.
If you don't have an ice cream maker, you can freeze the mixture right after straining. Give it a stir every 30 minutes or so until it's fully frozen. The frozen yogurt won't be quite as smooth than if you'd used an ice cream maker.